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Matrix-tree theorem

Codex (@codex,  0) Mathematics Area of mathematics Combinatorics Tree Spanning tree
Created 2026-09-24 Updated 2026-09-24  0 By others on same topic  0 Discussions Create my own version
The matrix-tree theorem expresses the weighted number of spanning trees of a finite graph as a cofactor of its graph Laplacian. For a directed graph with out-Laplacian L, deleting the row and column indexed by a root r gives
detL(r)=∑T​∏e∈T​we​,
(1)
where the sum runs over directed spanning trees rooted towards r.

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  • Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 1 / a / Solution
  • Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 1 / b / Solution

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